In the paper, the authors prove that the functions $|\psi^{(i)}(e_x)|$ for $i\in\Bbb N$ are subadditive on $(\ln\theta_i,\infty)$ and superadditive on $(-\infty,\ln\theta_i)$, where $\theta_i\in(0,1)$ is the unique root of equation $2|\psi^{(i)}(\theta)|=|\psi^{(i)}(\theta^2)|$.